Deviation (statistics)

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Not to be confused with Deviance (statistics).

In mathematics and statistics, deviation is a measure of difference between the observed value of a variable and some other value, often that variable's mean. The sign of the deviation (positive or negative), reports the direction of that difference (the deviation is positive when the observed value exceeds the reference value). The magnitude of the value indicates the size of the difference.

Types of deviation[edit]

Main article: Errors and residuals

A deviation that is a difference between an observed value and the true value of a quantity of interest (such as a population mean) is an error and a deviation that is the difference between the observed value and an estimate of the true value (such an estimate may be a sample mean) is a residual. These concepts are applicable for data at the interval and ratio levels of measurement.

Unsigned or absolute deviation[edit]

In statistics, the absolute deviation of an element of a data set is the absolute difference between that element and a given point. Typically the deviation is reckoned from the central value, being construed as some type of average, most often the median or sometimes the mean of the data set.

where

Di is the absolute deviation,
xi is the data element
and m(X) is the chosen measure of central tendency of the data set—sometimes the mean (), but most often the median.

Measures of deviation[edit]

Mean signed deviation[edit]

Main article: Mean signed deviation

For an unbiased estimator, the average of the signed deviations across the entire set of all observations from the unobserved population parameter value averages zero over an arbitrarily large number of samples. However, by construction the average of signed deviations of values from the sample mean value is always zero, though the average signed deviation from another measure of central tendency, such as the sample median, need not be zero.

Dispersion[edit]

Statistics of the distribution of deviations are used as measures of statistical dispersion.

Normalization[edit]

Further information: Normalization (statistics)

Deviations have units of the measurement scale (for instance, meters if measuring lengths). One can nondimensionalize in two ways.

One way is by dividing by a measure of scale (statistical dispersion), most often either the population standard deviation, in standardizing, or the sample standard deviation, in studentizing (e.g., Studentized residual).

One can scale instead by location, not dispersion: the formula for a percent deviation is the observed value minus accepted value divided by the accepted value multiplied by 100%.

See also[edit]